×

Higher-order and secondary Hochschild cohomology. (Cohomologie de Hochschild d’ordre supérieur et secondaire.) (English. French summary) Zbl 1360.13038

Higher-order Hochschild cohomology as introduced by T. Pirashvili [Ann. Sci. Éc. Norm. Supér. (4) 33, No. 2, 151–179 (2000; Zbl 0957.18004)] associates to a simplicial set \(X\), a commutative \(k\)-algebra \(A\) and an \(A\)-bimodule \(M\) the higher order Hochschild cohomology groups \(H^n_X (A,M)\). Secondary Hochschild cohomology as introduced by M. D. Staic [Algebr. Represent. Theory 19, No. 1, 47–56 (2016; Zbl 1345.16012)] associates with a triple \((A,B,\epsilon)\) (where \(\epsilon\) defines a \(B\)-algebra structure on \(A\)) and a \(B\)-symmetric \(A\)-bimodule the secondary Hochschild cohomology groups \(H^n \big( (A,B,\epsilon), M \big)\).
The two notions are unified in the present paper. The authors exhibit a construction that associates to a pair of simplicial sets \(Y \supseteq X\), a triple \((A,B,\epsilon)\) consisting of commutative \(k\)-algebras \(A\) and \(B\) and a \(k\)-algebra morphism \(\epsilon : A \rightarrow B\), and a symmetric \(A\)-bimodule \(M\) new cohomology groups denoted \(H^n_{(X,Y)} \big( (A,B,\epsilon), M \big)\). For \(X=Y\) this recovers higher-order Hochschild cohomology, for \(Y=D^2\) and \(X=S^1\) the construction yields secondary Hochschild cohomology.

MSC:

13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, D. W., Chain functors and homology theories, (Symposium on Algebraic Topology. Symposium on Algebraic Topology, Battelle Seattle Res. Center, Seattle, Wash., 1971. Symposium on Algebraic Topology. Symposium on Algebraic Topology, Battelle Seattle Res. Center, Seattle, Wash., 1971, Lect. Notes Math., vol. 249 (1971), Springer: Springer Berlin), 1-12 · Zbl 0229.55005
[2] Corrigan-Salter, B. R., Coefficients for higher order Hochschild cohomology, Homol. Homotopy Appl., 17, 1, 111-120 (2015) · Zbl 1321.13006
[3] Gerstenhaber, M., The cohomology structure of an associative ring, Ann. of Math. (2), 78, 267-288 (1963) · Zbl 0131.27302
[4] Gerstenhaber, M., On the deformation of rings and algebras, Ann. of Math. (2), 79, 59-103 (1964) · Zbl 0123.03101
[5] Gerstenhaber, M.; Schack, S. D., Algebraic cohomology and deformation theory, (Deformation Theory of Algebras and Structures and Applications. Deformation Theory of Algebras and Structures and Applications, Il Ciocco, 1986. Deformation Theory of Algebras and Structures and Applications. Deformation Theory of Algebras and Structures and Applications, Il Ciocco, 1986, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 247 (1988), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 11-264
[6] Ginot, G., Higher order Hochschild cohomology, C. R. Math. Acad. Sci. Paris, 346, 1-2, 5-10 (2008) · Zbl 1157.53042
[7] Ginot, G., Notes on factorization algebras, factorization homology and applications, (Mathematical Aspects of Quantum Field Theories. Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud. (2015), Springer: Springer Cham), 429-552 · Zbl 1315.81092
[8] Ginot, G.; Tradler, T.; Zeinalian, M., Higher Hochschild homology, topological chiral homology and factorization algebras, Commun. Math. Phys., 326, 3, 635-686 (2014) · Zbl 1334.13012
[10] Laudal, O. A., Formal Moduli of Algebraic Structures, Lect. Notes Math., vol. 754 (1979), Springer: Springer Berlin · Zbl 0438.14007
[11] Loday, J.-L., Cyclic Homology, Grundlehren Math. Wiss., vol. 301 (1992), Springer-Verlag: Springer-Verlag Berlin, appendix E by María O. Ronco · Zbl 0780.18009
[12] Pirashvili, T., Hodge decomposition for higher order Hochschild homology, Ann. Sci. Éc. Norm. Supér. (4), 33, 2, 151-179 (2000) · Zbl 0957.18004
[13] Staic, M. D., Secondary Hochschild cohomology, Algebr. Represent. Theory, 19, 1, 47-56 (2016) · Zbl 1345.16012
[14] Staic, M. D.; Stancu, A., Operations on the secondary Hochschild cohomology, Homol. Homotopy Appl., 17, 1, 129-146 (2015) · Zbl 1346.16004
[15] Yau, D., Deformation theory of modules, Commun. Algebra, 33, 7, 2351-2359 (2005) · Zbl 1135.16029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.